Graphics Reference
In-Depth Information
4
26
06
−
Now triangular so determinant is
product of main diagonal entries
=
3
00
−
−
1
=−
24.
An example application of determinants is Cramer's rule for solving small systems of
linear equations, described in the next section.
Determinants can also be evaluated through a process known as
expansion by
cofactors
. First define the
minor
to be the determinant
m
ij
of the matrix obtained by
deleting row
i
and column
j
from matrix
A
. The
cofactor c
ij
is then given by
1)
i
+
j
m
ij
.
c
ij
=
(
−
The determinant of A can now be expressed as
n
n
|
A
| =
a
rj
c
rj
=
a
ik
c
ik
,
j
=
1
i
=
1
where
r
and
k
correspond to an arbitrary row or column index. For example, given
the same matrix
A
as before,
|
A
|
can now be evaluated as, say,
4
−
2)
+
6
=−
4
26
250
−
50
1
20
25
|
A
| =
=
(
−
80
−
16
+
72
=−
24.
−
4
−
2
−
4
−
21
−
21
−
4
3.1.4
Solving Small Systems of Linear Equation Using
Cramer's Rule
Consider a system of two linear equations in the two unknowns
x
and
y
:
ax
+
by
=
e
, and
cx
+
dy
=
f
.